cspice_dvsep |
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## AbstractCSPICE_DVSEP calculates the time derivative of the separation angle between states. For important details concerning this module's function, please refer to the CSPICE routine dvsep_c. ## I/Os1 a double precision 6-vector defining a state; s1 = (r1, dr1 ). -- dt s2 a second state vector; s2 = (r2, dr2 ). -- dt An implicit assumption exists that 's1' and 's2' lie in the same reference frame with the same observer for the same epoch. If this is not the case, the numerical result has no meaning. the call: dvsep = ## ExamplesAny numerical results shown for this example may differ between platforms as the results depend on the SPICE kernels used as input and the machine specific arithmetic implementation. ;; ;; Load SPK, PCK, and LSK kernels, use a meta kernel for convenience. ;; cspice_furnsh, 'standard.tm' ;; ;; An arbitrary time. ;; BEGSTR = 'JAN 1 2009' cspice_str2et, BEGSTR, et ;; ;; Calculate the state vectors sun to Moon, sun to earth at ET. ;; cspice_spkezr, 'EARTH', et, 'J2000', 'NONE', 'SUN', statee, ltime cspice_spkezr, 'MOON', et, 'J2000', 'NONE', 'SUN', statem, ltime ;; ;; Calculate the time derivative of the angular separation of ;; the earth and Moon as seen from the sun at ET. ;; dsept = ## ParticularsIn this discussion, the notation < V1, V2 > indicates the dot product of vectors V1 and V2. The notation V1 x V2 indicates the cross product of vectors V1 and V2. To start out, note that we need consider only unit vectors, since the angular separation of any two non-zero vectors equals the angular separation of the corresponding unit vectors. Call these vectors U1 and U2; let their velocities be V1 and V2. For unit vectors having angular separation THETA the identity || U1 x U1 || = ||U1|| * ||U2|| * sin(THETA) (1) reduces to || U1 x U2 || = sin(THETA) (2) and the identity | < U1, U2 > | = || U1 || * || U2 || * cos(THETA) (3) reduces to | < U1, U2 > | = cos(THETA) (4) Since THETA is an angular separation, THETA is in the range 0 : Pi Then letting s be +1 if cos(THETA) > 0 and -1 if cos(THETA) < 0, we have for any value of THETA other than 0 or Pi 2 1/2 cos(THETA) = s * ( 1 - sin (THETA) ) (5) or 2 1/2 < U1, U2 > = s * ( 1 - sin (THETA) ) (6) At this point, for any value of THETA other than 0 or Pi, we can differentiate both sides with respect to time (T) to obtain 2 -1/2 < U1, V2 > + < V1, U2 > = s * (1/2)(1 - sin (THETA)) * (-2) sin(THETA)*cos(THETA) * d(THETA)/dT (7a) Using equation (5), and noting that s = 1/s, we can cancel the cosine terms on the right hand side -1 < U1, V2 > + < V1, U2 > = (1/2)(cos(THETA)) * (-2) sin(THETA)*cos(THETA) * d(THETA)/dT (7b) With (7b) reducing to < U1, V2 > + < V1, U2 > = - sin(THETA) * d(THETA)/dT (8) Using equation (2) and switching sides, we obtain || U1 x U2 || * d(THETA)/dT = - < U1, V2 > - < V1, U2 > (9) or, provided U1 and U2 are linearly independent, d(THETA)/dT = ( - < U1, V2 > - < V1, U2 > ) / ||U1 x U2|| (10) Note for times when U1 and U2 have angular separation 0 or Pi radians, the derivative of angular separation with respect to time doesn't exist. (Consider the graph of angular separation with respect to time; typically the graph is roughly v-shaped at the singular points.) ## Required ReadingICY.REQ ## Version-Icy Version 1.0.0, 07-DEC-2009, EDW (JPL) ## Index_Entriestime derivative of angular separation |

Wed Apr 5 17:58:00 2017